An interactive approach to bicriterion loading of a flexible assembly system

Modelling Vol. 25, No. 6, pp. 71-83, 1997 Copyright@1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 08957177/97 $17.00 + 0.00

Mathl. Comput.

Pergamon PII:

s0895-7177(97)00040-x

An Interactive Approach to Bicriterion Loading of a Flexible Assembly System T. J. Department

of Computer

University

SAWIK Integrated

of Mining

Faculty Al. Mickiewicza

Manufacturing

& Metallurgy

of Management 30, 30-059

Krakow,

Poland

[email protected] (Received

December

1996;

accepted

February

1997)

Abstract-This

paper presents integer programming formulations and an interactive solution procedure for a bicriterion loading problem in a flexible assembly system. The system is made up of a set of assembly stations linked with an automated material handling system. In the system, several different product types can be assembled simultaneously. The problem objective is to assign assembly tasks and products to stations with limited working space, so as to balance the station workloads and to minimize station-to-station product transfer time, subject to precedence relations among the tasks for a mix of product types. The solution procedure proposed is based on the weighting method and the interactive search for a set of weights which would produce the most preferred nondominated solution. Numerical examples are included to illustrate possible applications of the interactive approach for various problem formulations proposed. Keywords-Flexible ming.

assembly systems,

Production

planning,

Multiobjective

integer program-

NOMENCLATURE Indices

i

sssemblystation,i~I={l,...,m}

j

assemblytask,jE

k

producttype,kEK={l,...,p}

aij bi

amount of working space required for task j at station i total working space of station i (number of tasks that may be

dk

demand for product type k

J={l,...,n} Input

Parameters

assigned to station i, if all eij = 1)

%k

assembly time for task j and product type k

Qil

transportation

Ii

the set of stations capable of performing task j

Jk Rk

the set of immediate predecessor-successor

time from station i to station

1

the set of tasks required for product type k pairs of tasks (j, r) for

product type k such that task j must be performed immediately before task r

x

the weight factor in the objective

This work has been partially supported

by a KBN Grant.

71

function, 0 5 X 5 1

T. J. SAWIK

72

Decision Xij Yiljk

Variables

1, if task j is assigned to station i; otherwise xij = 0 1, if product type k after completion of task j on station i moves to station 1; otherwise l/i[$ = 0

&ljk

number of products

of type Ic to be transferred

from station i after

completion of task j, to station 1 to perform the next task

1. INTRODUCTION A flexible assembly system (FAS) is a set of assembly stations and a loading/unloading

(L/UL)

station linked with an automated material handling system (conveyors or automated guided vehicles), where different product types are assembled simultaneously. Each station consists of a single machine or of identical parallel machines, usually assembly robots. A robot has finite work space where a limited number of part feeders and gripper magazines can be placed. As a result only a limited number of tasks can be assigned to one assembly station, e.g., [l]. A typical assembly process proceeds as follows. A base part of an assembly is loaded on a pallet and enters the FAS at the L/UL station. As the pallet is carried by an automated guided vehicle (AGV) through assembly stations, components are assembled with the base part. When all the required components are assembled with the base part, it is carried back to the L/UL station and the complete assembly leaves the FAS. The assembly times are relatively small and usually shorter than vehicle transfer times. As a result, a stop-and-go AGV is often dedicated to each pallet from start of the assembly to finish. Therefore, transportation systems in FASs should have sufficient excess capacity to avoid bottlenecks in the system and its underutilization. Furthermore, when transportation times are greater than assembly times, neglecting the former at the machine loading level may lead to bottlenecks on some paths of the transportation network. Transportation times can also contribute to station idle time if stations have to wait for the delivery of the next product for assembly. The various issues involved with the design, loading, and scheduling of assembly systems have been extensively surveyed in [2], where the different aspects of the problem have been classified. According to this classification, the FAS is a mixed-model system, i.e., a system in which different models or products are assembled simultaneously. The two main loading objectives for such a system are to assign tasks to stations with limited working space in order to equalize station workloads and to minimize station-to-station product movements. For example, the objective of balancing workloads and minimizing station visits is used in [3] and minimizing product movements in [4]. These two criteria will usually conflict. Minimization of product movements may lead to workloads imbalance, that is unequal total assembly times assigned to stations and longer intermediate queues and bottlenecks in the system as a result. The less product movements required, the less material handling capacity that must be provided. In order to best utilize the system capabilities, the FAS loading problem should simultaneously account for both the processing times and the transfer times between the stations. One of the widely used modelling techniques employed to solve FAS design and loading problems is integer programming, e.g., [5-71. In [5,6], integer programs are presented to assign tasks to workstations and select assembly equipment in a multiproduct automated assembly system. A specialized branch and bound procedure based on general integer programming algorithm for assembly line balancing problem was developed in [7]. Most of the existing literature on short-term loading aspects has focused on a related loading problem in flexible manufacturing systems. Loading problem in flexible manufacturing systems involves assigning operations for selected part types and their associated tools to machines. Nonlinear mixed integer programming formulations and various loading objectives are suggested in [8]. Following the approach proposed in [8], [9,10] give formulations with bicriterion objectives. In 191,the objective of balancing workloads among machining centers and meeting due dates is

Flexible Assembly System

73

used, whereas the loading problem for part movement minimization and workload balancing is formulated in [lo]. The two common approaches used for solving the multiobjective integer programs are the lexicographic method and the weighting method, e.g., [ll]. In the former approach, the decision maker ranks the objective functions according to some preference order, and then sequentially solves the corresponding single objective optimization problems until a nondominated solution to the original multiple objective problem is obtained. In [12,13], sequential loading and routing procedures are proposed for solving the biobjective FAS loading problem. First, the station workloads are balanced using a linear relaxation-based heuristic, and then the best assembly routes to minimize total transportation time are selected based on a network flow model. The weighting approach is a common procedure to combine the objective

functions into a

single-objective function using a set of weights. The optimum solution to the resulting singleobjective problem is a nondominated solution to the original problem. There is no guarantee, however, that all efficient points are generated using the above methods. On the other hand, computing the set of all nondominated solutions to a multiple objective integer program clearly becomes unmanageable except for very small problems, e.g., [14]. In this paper, the bicriterion loading problem of a flexible assembly system is formulated as a biobjective integer program and an interactive solution procedure is proposed for finding a set of weights which would produce the most preferred nondominated solution. The paper is organized as follows. Integer programming formulations proposed for the bicriterion FAS loading are presented in Section 2. An interactive procedure to solve the problem heuristically is described in Section 3. Numerical examples are given in Section 4 and conclusions are made in the last section.

2. INTEGER

PROGRAMS

FOR A BICRITERION

FAS LOADING

The mathematical models presented in this section aim at simultaneous optimization of station workloads and station-to-station product flows. The problem objective is to determine an optimal assignment of tasks to stations and the corresponding assembly routes for all products so as to minimize station maximum workload and transportation times between stations. The models can be classified into the following two categories, e.g., [15]. 1. Models Ml, M2 for fixed assembly routes, where each task is assigned to only one station. 2. Models M3, M4 for alternative assembly routes, where each task can be assigned to more than one station. The models are constructed for a general FAS, in which revisiting of stations is allowed and assembly process of each product type is subject to individual precedence constraints. A feasible solution must satisfy the following three basic types of constraints. l l

l

The demand for all product types must be satisfied. The subset of tasks assigned to each station must not exceed the station working space available. Products must be routed to the stations where the required tasks have been assigned, subject to precedence relations.

In order to achieve the FAS loading objectives, one seeks to minimize the maximum station workloads, the maximum transfer time required to move products from one station to another, or the total transportation time required to transfer all products between the stations. The maximum workload P,,,, the maximum transfer time to a station Qmax, and the total transportation time QBUm,are defined below. (The notation used in the models is shown in the Nomenclature.)

T. J.

74

SAWIK

If only a single assembly route may be selected for each product type, the objective functions can be expressed ss follows:

(1)

(3) Otherwise, i.e., when more than one route may be selected for each product type, the objective functions are defined as below.

Pmax

(4

Q max

(5)

iEI

l#i

kEK jCJk

A bicriterion machine loading and part routing problem can be formulated by selecting either Qsum as a pair of the objective functions. The corresponding models

P max and Qmax or Pmax and Ml,

M3 and M2, M4 are presented below.

Model

Balancing station workloads and transfer times between the stations: fixed assembly

Ml.

routes. Minimize P max 7 &ma,

(7)

subject to

cc ccc

&PjkXij

I Pma~,

i

E

I,

(8)

5

i E

I,

69

j E

J,

(10)

kEK jf5.h

kEK

l#i

jEJk

dk9iiYiijk

Qmax,

c = c Xij

1,

iEIj

5 bi,

i E I,

yiljk

S 1,

/C E K,

i E Ij,

t! E I,,

1 #i,

(j,r)

E Rk,

(12)

Xlr + 2Yiljk

< 0,

k E K,

i E Ij,

1 E I,,

1 #i,

(j,r)

E .&c

(13)

aijxij

(11)

jEJ

Xcij + Xlr -Xij

-

Xij

E

{0,1}7

?/iljk E (0,

I},

V&j,

(14)

Vi, 1,j, k.

(15)

The objective function Pm= denotes total assembly time of the tasks assigned to the bottleneck station defined by constraint (8), and Qmax is the total transfer time required to deliver products to the station defined by constraint (9). Constraint (10) ensures that each task is assigned to

Flexible Assembly

System

75

only one station, and (11) is the staging capacity constraint.

Constraints (12) and (13) ensure

that each product type is routed to the stations where the required tasks are assigned. Implicitly, (12) and (13) account for the individual precedence constraints of each product type. Model

M2.

Balancing station workloads and minimizing total transportation time: fixed as-

sembly routes. Minimize P max, &sum,

(16)

subject to (3), (8), (lo)-(15). Model M3. Balancing station workloads and transfer times between the stations: alternative assembly routes. Minimize P max, Qmax,

(17)

subject to cc

iEIj

C

(xijk

&jk

=

dk,

k E K,

(j,r)

E Rk,

(18)

lEI, -

&rk)

=

0,

i E 1,

k E K,

(j,r)

E Rk,

(19)

1EI pjkyiljk

ccc kEK jEJk

1EI

ccc kEKjEJk

12%

I

qlixijk

c iEIi

xij

2

aijxij

c jEJ

I

Qmax,

1,

bi,

5

j

I,

(20)

i E I,

(21)

i E

Max,

J,

(22)

i E I,

(23)

E

yiljk

< dkxij,

kEKj

iEIj,

IEIr,

(j,r)ERk,

(24)

xljk

<

kEK,

iEIj,

LEI~,

(j,r)ERk,

(25)

dkxlr,

“ij xljk

2

E (07 I}, 0,

integer,

v’i,_i,

‘di,l,j,k.

(26)

(27)

Constraint (18) ensures that all required tasks be allocated among the stations. The number of products of type k assigned to station i to perform task j is the sum of appropriate product outflows Yiljk from station i after completion of task j, to all stations 1 E IT capable of performing the next task T such that (j, r) E Rk, i.e., ~1,1, Yiljk. Equality (19) are the network flow conservation equations for each station, product type, and a pair of successively performed tasks. Constraints (20) and (21) define the maximum workload and the maximum transfer time. Constraint (22) ensures that each task is assigned to at least one station, and (23) is the staging capacity constraint. Constraints (24) and (25) ensure that each product successively visits such stations where the required tasks may be assembled subject to precedence relations. Model

M4.

Balancing station workloads and minimizing total transportation time: alternative

assembly routes. Minimize P max, &sum,

(28)

subject to (6), (W, (W, (2% (2% (23), (24), (25) (2% (27). A common procedure to find efficient solutions to a multiobjective optimization problem which are not dominated by other solutions is to solve an equivalent single objective problem, where the original objective functions are combined into a single objective function using a set of weights.

T. J.

76

SAWIK

The weighting method reduces the original biobjective functions (7),(17) following single objective functions (29) and (30), respectively. W&+X + (I-

and (16),(28)

into the

4Qmam

(29)

~Pmax + (I- X)Qsum. Let us denote the reduced problems Ml,

M2, M3,

and M4X. The weight factor X E [0, l] can be interpreted

(30)

and M4, respectively,

MlX,

as a relative contribution

M2X,

M3X,

of assembly and

transportation times to the total completion time. However, such a contribution is not explicitly known beforehand. Therefore, a decision maker searches for a weight X which would produce the most preferred solution. Obviously, the optimal solution to the MU,. . . , M4X for any particular X > 0 would be an efficient solution to the corresponding bicriterion problem. A subset of strongly nondominated solutions can be generated by using the weights 0 < X < 1, otherwise weakly nondominated solutions are achieved.

3. AN INTERACTIVE

HEURISTIC

APPROACH

In this section, an interactive approach is presented to solve the bicriterion problems Ml, M2, M3, and M4 heuristically. The approach is based on searching for a weight X to the objective functions which would produce a solution most preferred by the decision maker, e.g., [16]. At each iteration, the decision maker further restricts the area of search in the set of all efficient solutions and guides the search along the efficient frontier for the most preferred solution. Briefly, the interactive approach proceeds as follows. STEP STEP

1. Set t = 1; t is the iteration number, t = 1,2,. . . . 2. In the X-range, select three test weights Xi, Xs, Xs, which allow for testing at one intermediate value, plus the end points for each X, and 1 - X,, p = 1,2,3. 3. Formulate three reduced single objective problems using the selected weights X,, p = 1,2,3. 4. Solve each of the reduced single objective problems using a suitable algorithm (optimal or a heuristic). 5. From the resulting solutions (task assignments and assembly routes), select the one most preferred based on the decision maker’s preferences or a computer simulation of the corresponding detailed assembly schedules. For example, select the solution leading to the schedule with the shortest length. 6. In the X-range under investigation, could there still be values of X which might give a more preferred solution. If yes, set t = t + 1 and go to STEP 2. If no, stop.

STEP STEP STEP

STEP

In the first iteration, three test weights Xi = E, X2 = 1 -E, and Xa = l/2, uniformly distributed in the original X-range (0,l) are chosen, where E is a sufficiently small positive number. The search range decreases in the subsequent iterations. The search is concentrated in a region surrounding the weight X, which produced the preferred solution in the previous iteration. The new search region is obtained using a contraction mechanism in such a way that it contains the weight X, which produced the preferred solution. The new three test weights are chosen such that they are uniformly distributed over the contracted X-range. The recursive relationships between weights Xi+,+’and Xk (p = 1,2,3) in the iterations t + 1 and t are given below. f At+’ P

= f

x;,

if the preferred solution in iteration t was obtained for Xt E (0,1/2),

(1+X;),

ifth e p re ferred solution in iteration t was obtained for Xt E (l/2,1)

A;,

,

if the preferred solution in iteration t was obtained for Xt = l/2. (31)

Flexible Assembly System

77

The procedure stops if there are no new solutions or when the search region becomes small enough for it to contain no new solutions which might be preferred by the decision maker. In STEP 4, either an optimal algorithm like a branch and bound or a heuristic can be used, especially for large-size problems. Using a heuristic does not guarantee that an optimal solution to the reduced single objective problem would be obtained. Instead, it is possible that a nearefficient solution to the bicriterion problem would be found which could be less preferred than a corresponding

unobtained point. However, by solving several reduced single objective problems

within the same X-range, this efficient point can be uncovered even if a heuristic is used, e.g., [16].

4. COMPUTATIONAL

EXAMPLES

In this section, several numerical examples are presented to illustrate various applications of the models proposed. The models have been applied for loading of a hypothetical FAS made up of m = 3 identical stations, in which n = 15 task types are assembled to produce p = 4 product types.

The corresponding

ordered sequences of tasks j E Jk required to make each product

type k = 1,2,3,4 are the following: (1,2,3,4,6,12,14,15), (1,2,5,6,9,10,13,15), (2,4,5,7,8,9,10,14), (8,11,13,14,15). Each product type k has the same demand of dk = 100 units. The system is organized as a series of the stations 1,2,3 located along a bidirectional AGV guide path. The transfer time between the successive stations is equal to 2 time units, and hence, transportation times qil from station i to station 1 are q12 = q2l = 2, 413 = 431 = 4, q-23 = 432 = 2. Processing times pjk for each task j are identical for all product types (i.e., pjk = pj,%, j E Jk) and are as follows: pi = 4, p2 = 2, p3 = 2, p, = 2, p5 = 4, ps = 2, p7 = 3, ps = 5, pg = 2, 4, Pll = 5, P12 = ‘A P13 = 4, P14 = 2, P15 = 3. The components are assumed to be all of relatively similar sizes so that each part type feeder uses the same amount of a station finite working space. As a result, one can substitute aij = 1 for all i,j in the staging capacity constraints (11),(23).

PlO =

Fixed

Assembly

Routes

Models MlX and M2X with various values of the weight X E {0.00,0.05,. . . ,0.95,1.00} have been applied to determine task assignments and single assembly routes for a FAS, in which each station i, (i = 1,2,3) has bi = 5 part feeders. Simple lower bounds on the maximum workload P,,,,,, the maximum transfer time Q,,,, and the total transfer time Q,,, for the example are LBP,,,

=

(32)

where (Pi,..., @‘m denotes a partition of the set (1,. . .p} of p product types into m disjoint subsets, and [u] is the smallest integer not less than a. The three variants of the loading problem with fixed assembly routes have been considered in which the objective function accounts for: transfer times only (X = 0), both assembly and transfer times (0.05 5 X 5 0.95), and assembly times only (X = 1). The solution results for a are shown in Tables 1 and 2, respectively. The only pair of criteria Pm,,, Q,, and Pm,, Q,,, solution obtained for X E [0.05,0.85] is identical for both pairs of criteria. This is a result of predominated constraints (8),(g) or (8),(3) for the example data. For a comparison, problems MlX and M2A have also been solved for the csse of unlimited staging capacities, i.e., assuming that bi = n for all i = 1,2,3. For X E [0.05,0.85], the same

T. J.

78 nble 1. Task routes.

x

&lax,

0

Station 1

3400,800

[0.05,0.85]

assignmentsand assemblyroutes for Pmax and Q,,,= criteria: fixed Task Assignments

Qmax

*3006, 800*

SAWIK

Station 2

(*)

Routes cb) for Product 2

1

Station 3

Types 4

3

1,2,3

12 f t12, 93

2,3

12347 1t >,

568910 I , I ,

11,12,13,14,15

1,2,3

1,2,3,4,5

678910 , , , ,

11,12,13,14,15

1,2,3

1,2,3,1

1,2,3

2,3

1,2,3

2,3

232321 I >1t ,

2,1,3

[0.90,0.95]

2900, 1000

1,2,4,5,7

3,6,8,11,13

9,10,12,14,15

1,2,1,2,3

1,2,3,2,3

1

2900, 2000

1,11,12,13,14

2,3,7,8,10

4,5,6,9,15

12313 $10

123213 ,,I,,

(a) subsets of assigned tasks j (b) sequences of successively

visited stations i

* * the most preferred solution Table 2. Task assignments and assembly routes for P,, routes.

and QeUm criteria:

fixed

(a) subsets of assigned tasks j (b)

sequencesof successivefyvisited stations i

* * the most preferred solution

values of the objective functions P,,,,, = 3000, Qmax = 800, and Qsum = 1400 have been obtained. Such an insensitivity to the problem parameters has not been observed for the other values of weight X. The solutions obtained for X = 0 and X = 1 are weakly nondominated, whereas the remaining solutions are strongly nondominated. Notice that there exist alternate optima for the example, e.g., exchange of task assignments between stations 1 and 3 does not change values of the objective functions. The integer programs MlX and M2X for the example are composed of 323 constraints with 195 variables. Alternative

Assembly

Routes

The interactive procedure presented in Section 3 and models M3X, M4X have been applied to optimally balancing of a FAS with alternative assembly routes, where each station i, (i = 1,2,3) has bi = 8 part feeders. The lower bounds (33) and (34) on the maximum and the total transfer time are LL3Qmax = 200 and LBQ,,, = 200, respectively. In the first iteration, the optimal task assignments and assembly routes have been determined for weights Xi = E = 0.05, X; = 1 - E = 0.95, X; = 0.50. For model M3X, the solution for Xi = 0.50 with Pm,, = 2900 and Qmax = 400 is selected to be the most preferred solution. Therefore, the interactive search terminates at the first iteration, since Y+i = Xt, see (31). For the best solution, the selected assembly routes are determined by the following set of flow variables Y&k: G,1,6,3

=

40,

K,1,13,4

=

10%

%,2,7,3

=

&,3,1,2

yl,1,6,4

= 100,

G&9,3

= 100,

=

10%

&,1,7,3

=

40,

&,1,8,3

=

6%

6%

y2,2,9,2 =

100s

%,2,10,2

=

100,

%,2,13,2

=

=

100,

y3,2,5,2 =

100,

&,2,5,3

=

100,

&,3,2,1

&,3,2,3

=

10%

y3,3,3,1 =

100,

&,3,4,x

=

100,

&,3,12,1

=

100,

K&3,14,1 =

100.

K&14,4

&,1,10,3

= 100,

&,11,4

=

100,

ti,2,6,2

=

100,

10%

%,3,1,1

=

10%

=

10%

&,3,2,2

=

10%

y3,3,4,3 =

100,

&,3,6,1

=

10%

FlexibleAssemblySystem

79

For model M4X, the most preferred nondominated solution with P,,,,, = 2900 and Qsum = 460 was obtained for Xi = 0.95, and hence, in the second iteration M4X was solved for three new weights (see, (31)) Xf = (l/2)(1 + Xi) = 0.525, Xg = (l/2)(1 + Xf) = 0.975, Xi = (l/2)(1 + Xi) = 0.750. A new solution with Pm, = 2900 and Qsum = 728 was found only for X = 0.750, whereas the other solutions are identical with those obtained in the first iteration. The new solution is dominated by the best solution found in the first iteration. Therefore, the search procedure stops and the optimal solution for X = 0.95 is selected to be the best solution to the problem. optimal assembly routes are defined by the following set of variables Y&k: y1,1,2,3

=

100,

y1,1,9,3

=

100,

y1,1,4,3

Y2,2,ti,l = 100, &,2,14,1

=

100,

K,1,5,3

=

10%

y1,1,7,3

=

100,

&,1,8,3

=

100,

K,1,10,3

=

100,

Yl,2,8,4

=

100,

%,2,3,1

=

10%

r,,2,4,1

=

100,

%,2,11,4

=

100,

&,2,12,1

=

100,

&,2,13,2

=

2%

&,2,13,4

=

100,

%,3,14,4

=

1,

&,2,2,1

=

100,

fi,2,10,2

=

29,

&,3,5,2

=

100,

&,3,6,2

=

100,

=

10%

&,2,14,4

=

9%

y3,3,1,1

=

100,

&,3,1,2

=

100,

%,3,2,2

=

10%

%,3,5(,2

=

100,

&,3,10,2

=

71,

&,3,13,2

=

71.

The

The task assignments obtained for models M3X and M4X using the interactive procedure are shown in Tables 3 and 4, respectively. The results presented above for the best solutions indicate that model M3X selected two alternative assembly routes for product type 3, whereas model M4X selected two alternative routes for each product type 2 and 4. The assembly routes for the other solutions are not presented. The integer programs M3X and M4X for the example are composed of 564 constraints with 282 variables. Table 3. Task assignments for Pmax and Q max criteria: alternative routes.

x

f’ max,

Task Assignments

Qmax

Station 1

Station 2

Station 3

0.05

3250, 300

2,4,8,10,11,13,14,15

1,2 ,3 ,4 ,5,6,9,12

0.50

*2900, 400*

4,8,9,10,11,13,14,15

1,6 ,7 ,8 ,9,10,13,15

2 I3 ,4 >5 16 I 12,14,15

0.95

2900, 658

1,2,3,4,5,6,13,14

1,5,7,9,10,13,14,15

8,9,10,11,12,13,14,15

Table 4. Task assignments for Pmax and Q sum criteria:

2 >5 ,7 ,8 19 I 10,14,15

alternative routes.

Task Assignments

x

P m&x>Qs,,

0.050

3300, 400

1 ,2 >3 ,4 >6 112,14,15

5,6,9,10,11,13,14,15

2,4 ,5 97 ,8 ,9 , lo,14

0.500, 0.525

3100, 400

1,2,5,6,9,10,13,15

3,4,6,11,12,13,14,15

2,4,5,7,8,9,10,14

0.750

2900, 728

1,2,3,4,5,6,7,8

4,6,9,10,12,13,14,15

7,8,9,10,11,13,14,15

0.950, 0.975

*2900, 460*

2,4,5,7,8,9,10,14

3,4,6,11,12,13,14,15

1,2,5,6,9,10,13,15

The Effect

of Varying

Demand

Station 1

Station 2

Station 3

for Products

The effect of varying demand for products under fixed and flexible routing was investigated assuming that demand dr for product type 1 suddenly increased from 100 to 200 units, all things being equal. Table 5 and 6 compare solution results obtained for a pair of criteria Pm, and Q,,,, for fixed and alternative assembly routes, respectively. The lower bounds (32) and (33) for the modified example data are LBP,, = 3534and LB&,*, = 400. For the fixed routing model MlX, the solution with P,,,,= 3600 and Q,, = 1200 obtained for Xi = 0.95 in the first iteration of the interactive heuristic dominates the other solutions, and hence, in the second iteration MlX was solved for weights X: = 0.525, Xi = 0.975, Xi = 0.750. The only new solution with P,,,,, = 3700 and QmBX= 1000 obtained for X = 0.750 was selected

T. J. SAWIK

80

Table 5. Task assignments for dl = 200: fixed routes.

x

P max, Qmax

0.0500

4000. 1200

0.5000

1 3800, 1400 1 3600. 1600

0.5250

Task Assignments Station 2

Station 3

1 5.6.9.11.120 1 6,7,12,14,15 I 12.13.14.15

1 7.10.13.14.15 1 8,9,10,11,13 I 1.2.5.7.11

Station 1 1.2.3.4.8

1 1,2,3,4,5 1 4.6.8.9.10

0.7500

*3700, 1000*

1,2,3,4,7

5,6,8,9,10

11,12,13,14,15

0.7625, 0.8750

3600, 1400

1,2,3,5,7

4,6,12,14,15

8,9,10,11,13

0.9500

3600. 1200

0.9750,0.9875

1

3600, 2000

1.2.3.4.6

1 1,2,3,5,12

Table 6. Task Assignments x 0.050

P max,

&n-tax

4200, 400

5.8.9.10.12

7.11.13.14.15

1 4,6,7,14,15

1 8,9,10,11,13

for dl = 200: alternative routes. Task assignments

Station 1

Station 2

Station 3

2,4,6,8,11,13,14,15

2,4,5,6,7,9,10,14

1 ,2 13 ,4 ,6 t 12,14,15

0.500, 0.525

3700, 800

2,3,4,6,10,12,14,15

1,2,4,6,8,9,13,14

2 94 ,5 T6 ,7 , 11,14,15

0.750, 0.950

*3534, 734*

2,4,6,10,11,13,14,15

124568914 I 17I 1,

2 >3 ,4 ,6 , 7 , 12,14,15

0.975

3534, 800

2,4,5,6,7,8,14,15

1 ,2 ,4 16 79 111,13,14

,

2 >3 ,4 ,6 ,10,12,14,15

Table 7. Task assignments for min qil = rnaxpjk = 5: fixed routes.

be the most preferred. In the third iteration, MlX was solved for Xi = 0.7625, A$ = 0.9875, Xi = 0.875, however, the new solutions found are less preferred than that for X = 0.750. For the alternative routing model M3X, the solution with Pm,, = 3534 and Q,, = 734 obtained for Xi = 0.95 is selected in the first iteration. Then, in the second iteration M3X was solved for three new weights XT = 0.525, Xg = 0.975, AZ = 0.750. However, the only new solution obtained for A% = 0.975 is dominated by that selected in the first, iteration. For simplicity, the assembly routes selected are not presented. Comparison of the most preferred solutions selected under fixed and flexible routing shows that if the demand for product type 1 is increased from 100 to 200 units, the maximum workload Pnmx will be increased from 3000 to 3700 for fixed routing and from 2900 to 3534 for flexible routing, whereas the maximum transfer time Qmax will be increased from 800 to 1000 and from 400 to 734, respectively, for fixed and flexible routing. to

The Effect of Varying Transportation

Time

The effect of varying spacing, and therefore, transportation time was investigated assuming that the transfer time between successive stations is equal to the longest assembly time, that is to 5 time units, all things being equal. Hence, the transportation times qil from station i to station 1 are q12 = 421 = 5, q13 = 431 = 10, 423 = 432 = 5. Table 8. Task assignments for min I& = mapjk

X

Pmax, Qmn

0.05

4500, 2000

0.50 0.95

= 5: alternative routes.

Task Assignments Station 1

Station 2

Station 3 2 t3 *4 ,5 ,6 ,12,14,15

1,2,4,5,6,7,14,15

2,8,9,10,11,13,14,15

*3050,2050*

2,4,8,9,10,13,14,15

2 ,4 I5 ,6 I7 , 12,14,15

1,2,3,8,11,13,14,15

2900, 3000

2 I4 I5 I7 ,8 I 11,14,15

2,4,5,9,10,13,14,15

112 I3 ,4 I6 I12,14,15

Flexible Assembly

System

Tables 7 and 8 compare solution results obtained for a pair of criteria P,,, and alternative assembly routes, respectively. datais LBQ,,, = 2000.

81

and Qsum, for fixed

The lower bound (34) for the modified example

For the fixed routing model M2X, in the first iteration of the interactive heuristic, the same solution with Pm,, = 3000 and Qs,, = 3500 was obtained for all three test weights Xi = 0.05, Xi = 0.95, Xi = 0.50. Therefore, the procedure terminates at the first iteration. For the alternative routing model M4X, the solution for Xi = 0.50 with Pmax = 3050 and Q sum - 2050 is selected to be the most preferred solution. Therefore, the interactive search also terminates a.t the first iteration, since X‘+’ = Xt, see (31). The selected assembly routes are not presented. Comparison of the most preferred solutions selected under fixed and flexible routing shows that if the transfer time between successive stations increased from 2 to 5 time units, the maximum workload P,,, will remain unchanged at 3000 for fixed routing and will increase from 2900 to 3050 for flexible routing, whereas the total transportation time Qs,, will increase from 1400 to 3500 and from 460 to 2050, respectively, for fixed and flexible routing. The above results have indicated that the flexible routing policy is capable of better accommodating the varying process parameters and yields lower values of the objective functions. Computational

Experiments

In order to evaluate the effectiveness of the approach proposed for FAS loading, 50 test problems have been solved using the interactive search procedure and models MlX, M2X, M3X, M&L The test examples were constructed for a FAS with 3 < m 5 5 assembly stations each having 3 5 bi 5 10 part feeders all of similar sizes (i.e., all aij = l), in which 10 < n 5 40 task types are required to simultaneously assembly 2 5 p 5 5 product types, where each product type k requires 5 5 IJk] I 15 different assembly tasks. The assembly and transportation times pjk and qil were uniformly distributed over [l,lO], and the demands dk for product types were uniformly distributed over [25,100]. The interactive procedure was terminated if there were no new solutions found. For such a termination rule, the number of iterations of the interactive heuristic for the test problems was not greater than three. The computational experiments have indicated that there exist many alternate optima that yield the sarne values of the objective functions for different task assignments, different assembly routes and weights X. On the other hand, if alternative routes are allowed, the same task assignment with different assembly routes and different values of the objective functions can be obtained for different X. The maximum workload Pm,, more often reaches its lower bound LBP,,, (32)for flexible routing as well as the other objectives functions take on lower values than those for fixed routing, all things being equal. The experiments have also indicated that the flexible routing models M3X and M4X are much more sensitive to the varying relative importance between pairs of criteria, than models MU and M2X for the fixed routing. If only single assembly routes are available, the optimal routes are slightly sensitive to the value of selected weight X, while they are much more dependent on X, when alternative routes are allowed. In general, the tighter are staging capacity constraints, and the closer values of assembly and transportation times, the more indifferent to the weight X is the solution obtained using the interactive heuristic. For many test examples [17], the weight X of the best solution was observed to reflect relative contribution of the two criteria to the value of the reduced single objective function obtained. The value of X was found to be close to the ratio of Pm, /(Pmax+Qmax) and pm,l(Pm,x+Qsum)7 respectively, for (29) and (30). Such values of X were observed particularly in the case of fixed routing. The above results suggest some possible reduction of the computations required by the interactive search procedure. At each iteration t 2 2 of the heuristic, instead of three reduced

T. J. SAWIK

82

single objective problems formulated for three weights X (31), one would need to solve only one reduced problem formulated for Xt = l/(1 + Qt-‘/Pt-‘), where ptml and Qt-’ denote values of the two objective functions for the most preferred solution selected at iteration t - 1. However, such a modification of the search procedure may further restrict the subset of efficient solutions generated using this approach. The examples presented in this section and the test problems were solved using discrete optimizer LINGO [18], which permits a compact problem specification using a mathematical modelling language. CPU run times on a PC 486 have been varied from few seconds to several minutes. The experiments have indicated that the computation time increases with - the number m of stations, n of task types, and p of product types; - the slackness of the system working space, e.g., measured by the ratio (x7

bi - n)/n.

The CPU time required for solving M3X and M4X was much larger than that for MlX and M2X, and in particular M4X required the largest computation time. 5.

CONCLUSION

The models and the interactive solution procedure proposed may be used as a basis of a simple decision support tool for short-term planning and scheduling in flexible assembly systems, where workloads of both the assembly stations and the material handling facilities should be balanced simultaneously. A multiobjective integer programming seems to be an appropriate approach to the FAS loading problem. The routing variables representing station-testation

movements of products contribute essen-

tially to the problem size, especially when alternative assembly routes are allowed. On the other hand, a typical FAS includes only a few quite versatile assembly stations, where only a subset of all required components is assembled simultaneously. For example, in mechanical assembly smaller subsets of parts are first assembled into several subassemblies which are next assembled into the final products. In such cases, the models proposed can be solved even by commercially available codes. The results of computational experiments have indicated that for the loading problems considered may exist many alternate optima with the same values of the objective functions for different task assignments and assembly routes, in particular under flexible routing policy. Comparison of the results obtained for flexible and fixed routing indicates that the flexible routing policy leads to smaller values of the loading criteria selected, and hence, to better utilization of the system capabilities. REFERENCES 1. H.F. Lee and R.V. Johnson, A line-balancing strategy for designing flexible assembly systems, International Journal of Flezib~e Manufucturing Systems 3 (2), 91-120 (1991). 2. S. Ghosh and R.J. Gagnon, A comprehensive literature review and analysis of the design, balancing and scheduling of assembly systems, International Journal of Production Research 27 (4), 637-670 (1989). 3. J.C. Ammons, C.B. Lofgren and L.F. McGinnis, A large scale machine loading problem in flexible assembly, Annals of Operations Research 3, 319-332 (1985). 4. A. Agnetis, C. Arbib, M. Lucertini and F. Nicolo, Part routing in flexible assembly systems, IEEE Inaneaction on Robotics and Automation 6, 697-705 (1990). 5. SC. Graves and B.W. Lamar, An integer programming procedure for assembly system design problems, Operations Research 31 (3), 522-545 (1983). 6. S.C. Graves and C.H. Redfield, Equipment selection and task assignment for multiproduct assembly system design, International Journal of Flexible Manufacturing Systems 1 (I), 31-50 (1988). 7. F.B. Talbot and J.H. Paterson, An integer programming algorithm with network cuts for solving the assembly line balancing problem, Management Science 30 (l), 85-99 (1984). 8. K.E. Stecke, Formulation and solution of nonlinear integer production planning problems for flexible manufacturing systems, Management Science 29, 283-288 (1983).

Flexible Assembly

System

83

9. K. Shanker and Y.-J.J. Tzen, A loading and dispatching problem in a random flexible manufacturing syst,em, International Journal of Production Research 23 (3), 579-595 (1985). 10. B.K. Modi and K. Shanker, A formulation and solution methodology for part movement minimization and workload balancing at loading decisions in FMS, International Jounzal of Production Economics 34 (1) (1994). 11. F. Seidarovszky, M.E. Gershon and L. Duckstein, Techniques for Multiobjective Decision Making in Systems Management, Elsevier, Amsterdam, (1986). 12. T. Sawik, Sequential loading and routing in a FAS, In Proceedings of Rensselaer’s Fifth International Conference on Computer Integrated Manufacturing & Automation Technology, Grenoble, May 29-31, 1996, pp. 48-53. 13. T. Sawik, A two-level heuristic for machine loading and assembly routing in a FAS, In Proceedings of the ph IEEE International Conference on Emerging Technologies and Factory Avtomution, Kauai, HI, November 18-21, 1996, pp. 143-149. 14. G.R. Bitran, Theory and algorithms for linear multiple objective programs with zero-one variables, Adathematical Programming 17, 362-390 (1979). 15. T. Sawik, Integer programming models for the design and balancing of flexible assembly systems, Mat/d. Comput. Modelling 21 (4), 1-12 (1995). 16. D. Gabbani and M. Magazine, An interactive heuristic approach for multi-objective integer-programming problems, Journal of Operational Research Society 37, 285-291 (1986). 17. T. Sawik, Planowanie i sterowanie produkcji w elastycznych systemach produkcyjnych (Production Planning and Control in Flexible Assembly Systems), (in Polish with English summary), WNT Publishers, Warszawa, (1996). 18. L. Schrage and K. Cunningham, LINGO Optimization Modeling Language, LINDO Systems, Chicago, (1991).